In this talk, we discuss two
methods used in the modelling of materials with defects at different scales.
The first, known as the method
of mesoscale asymptotic approximations [1], is appropriate for the
modelling of granular materials. In particular, we discuss the approximation of
solutions to a particular class of transmission problems in solids containing
clusters of many small inclusions, which can interact with each other [2]. The
asymptotics are supplied with remainder estimates that are rigorously
justified.
The second method concerns the
dynamic failure of structured materials, commonly used in understanding the
phenomena involved at multiple scales in the fracture of solids. The subject
has been explored in the last 30 years for a variety of mass-spring systems
[3].
Here, we take a different
approach and consider faults propagating through periodic mass-beam systems [4,
5], which are more commonly found in civil engineering applications such as
long rooftops, bridges and pipeline systems. We give a summary of analytical
results, based on the Fourier transform and the Wiener-Hopf technique,
concerning the dynamic behaviour of the structure during the failure.
Numerical simulations
demonstrate the effectiveness of both approaches.
It is envisaged that both techniques
may find a new home in the modelling of phenomena associated with the behaviour
of sea ice.
References:
[1] V. Maz’ya, A. Movchan and
M. Nieves, (2013): Green’s Kernels and Meso-Scale Approximations in Perforated
Domains, Lecture Notes in Mathematics 2077, Springer.
[2] M.J. Nieves, (2017):
Asymptotic analysis of solutions to transmission problems in solids with many
inclusions, SIAM J. Appl. Math. 77 (4), 1417-1443.
[3] L.I. Slepyan,
(2002): Models and Phenomena in Fracture Mechanics, Foundations of
Engineering Mechanics, Springer.
[4] M.J. Nieves, G.S.
Mishuris, L.I. Slepyan, (2016): Analysis of dynamic damage propagation in
discrete beam structures, Int. J. Solids Struct. 97-98, 699-713.
[5] M.J. Nieves, G.S. Mishuris, L.I. Slepyan,
(2017): Transient wave in a transformable periodic flexural structure, Int. J.
Solids Struct. 112, 185-208